# Lecture 7 - 7 Boundary Value Problems for ODEs Boundary...

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7 Boundary Value Problems for ODEs Boundary value problems for ODEs are not covered in the textbook. We discuss this important subject in the scalar case (single equation) only. 7.1 Boundary Value Problems: Theory We now consider second-order boundary value problems of the general form y ( t ) = f ( t, y ( t ) , y ( t )) a 0 y ( a ) + a 1 y ( a ) = α, b 0 y ( b ) + b 1 y ( b ) = β. (55) Remark 1. Note that this kind of problem can no longer be converted to a system of two first order initial value problems as we have been doing thus far. 2. Boundary value problems of this kind arise in many applications, e.g., in me- chanics (bending of an elastic beam), fluids (flow through pipes, laminar flow in a channel, flow through porous media), or electrostatics. The mathematical theory for boundary value problems is more complicated (and less well known) than for initial value problems. Therefore, we present a version of an existence and uniqueness theorem for the general problem (55). Theorem 7.1 Suppose f in (55) is continuous on the domain D = { ( t, y, z ) : a t b, -∞ < y < , -∞ < z < ∞} and that the partial derivatives f y and f z are also continuous on D . If 1. f y ( t, y, z ) > 0 for all ( t, y, z ) D , 2. there exists a constant M such that | f z ( t, y, z ) | ≤ M for all ( t, y, z ) D , and 3. a 0 a 1 0 , b 0 b 1 0 , and | a 0 | + | b 0 | > 0 , | a 0 | + | a 1 | > 0 , | b 0 | + | b 1 | > 0 , then the boundary value problem (55) has a unique solution. Example Consider the BVP y ( t ) + e - ty ( t ) + sin y ( t ) = 0 , 1 t 2 , y (1) = y (2) = 0 . To apply Theorem 7.1 we identify f ( t, y, z ) = - e - ty - sin z . Then f y ( t, y, z ) = te - ty which is positive for all t > 0, y, z R . So, in particular it is positive for 1 t 2. Moreover, we identify f z ( t, y, z ) = - cos z , so that | f z ( t, y, z ) | = | - cos z | ≤ 1 = M. Obviously, all continuity requirements are satisfied. Finally, we have a 0 = b 0 = 1 and a 1 = b 1 = 0, so that the third condition is also satisfied. Therefore, the given problem has a unique solution. 78

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If the boundary value problem (55) takes the special form y ( t ) = u ( t ) + v ( t ) y ( t ) + w ( t ) y ( t ) y ( a ) = α, y ( b ) = β, (56) then it is called linear . In this case Theorem 7.1 simplifies considerably. Theorem 7.2 If u, v, w in (56) are continuous and v ( t ) > 0 on [ a, b ] , then the linear boundary value problem (56) has a unique solution. Remark A classical reference for the numerical solution of two-point BVPs is the book “Numerical Methods for Two-Point Boundary Value Problems” by H. B. Keller (1968). A modern reference is “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations” by Ascher, Mattheij, and Russell (1995). 7.2 Boundary Value Problems: Shooting Methods One of the most popular, and simplest strategies to apply for the solution of two-point boundary value problems is to convert them to sequences of initial value problems , and then use the techniques developed for those methods. We now restrict our discussion to BVPs of the form y ( t ) = f ( t, y ( t ) , y ( t )) y ( a ) = α, y ( b ) = β. (57) With some modifications the methods discussed below can also be applied to the more general problem (55).
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